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Forcing is a method first used to prove the continuum hypothesis is independent of the classical axioms of set theory

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Does ZF + BPI alone prove the equivalence between "Baire theorem for compact Hausdorff space...

Rasiowa-Sikorski Lemma (for forcing posets)is the statement: For any p.o. $\mathbb{P}$ (i.e. … Does any of you know if it is indeed possible to prove the equivalence between "Baire Category Theorem for Compact Hausdorff Spaces" and "Rasiowa-Sikorski Lemma for forcing posets" from ZF + BPI alone …
Samuel G. Silva's user avatar